If i am at point a and want to go to point B at half the distance with each move when will i get there?

Question

If i am at pointA and want to go to point B at half the distance with each move when will i get there? or will I ever get there?
Is there a mathmatical equation for this?

Answer 1

This is one of the famous paradoxes of Zeno, the Greek philosopher, who used it to argue that motion is impossible. You get to point B, but only after an infinite number of moves. Let the distance from A to B be 1. After N moves, you are a distance 1/(2^n) away from B. E.g., after 1 move, you are 1/2 unit from B; after 2 moves you are 1/4 unit from B; after 10 moves you are 1/1024 unit from B, and so on. The sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .. . 1/2^n . . . equals 1.

If each move takes the same time, then you never get to B during any finite time. But (as pointed out above) if each move takes half the time as the move before (i.e., if you are moving at a constant speed) then you get to B in a finite time even though you make an infinite number of moves.

Answer 2

No, strictly speaking you'll NEVER get there IF you take the SAME TIME for each move.

Look at it this way. Let the distance from A to B be ' D.' Suppose you move half (1/2) the distance D in one unit of time, t. (It could be a second, a minute, whatever.) You're still 1/2 D away. Now move half of THAT distance again, but still in the same time ' t .' Now you've taken time 2t, but you're still 1/2 of 1/2 D away, that is (1/2)^2 D away.

You can see where we're heading! After ' n ' moves, it will have taken you a total time of ' nt ,' but you'll STILL be (1/2)^n D away from your goal. Of course, you could say that that is NEGLIGIBLE after ' n ' gets sufficiently large, but NEVERTHELESS, strictly speaking, you'll NEVER arrive.


The situation becames MARKEDLY DIFFERENT, however, if you change just one thing. Now maintain the SAME SPEED for each move. Let that speed v be such that your original step of (1/2) D takes you time t, so that v = (1/2) D / t. You've (1/2) D to go, and have taken time ' t ' so far. In the second move, you again go half the remaining distance, so that at the end of the move you have both gone, and have left to go, exactly the same distance, (1/2)^2 D or 1/4 D. At your MAINTAINED speed of v = (1/2) D / t, that step took you a time of (1/4) D / v = (1/4) D / [(1/2) D / t] = (1/2) t, when the algebraic dust has settled.

Guess where we're now going with this: after the next step, you've still got (1/2)^3 D or (1/8) D to go, and it took you (1/4) t, etc.

So you've still got a GEOMETRICALLY DECREASING distance to go at any stage, apparently exactly as before. HOWEVER, the elapsed time is now COMPLETELY DIFFERENT! The time taken has been t (1 + (1/2) + (1/2)^2 + (1/2)^3 + (1/2)^4 + ... ) and THAT CONVERGES, to precisely 2t, unlike the first case in which the time tended to infinity!

(2t is of course OBVIOUSLY correct in this second case, because we said the whole trip would be done at a constant speed. And of course, if it took time ' t ' to go HALF the distance, it will take time ' 2t ' to go the FULL distance. Duuuu!)

So the answer is that whether you get there or not IN A FINITE TIME depends on more than whether you mentally divide up the journey into successive "halves"; it also depends on things like the speed or the time intervals employed beween or during each step or move.

An analogous mathematical analysis lies behind the explantion/understanding of Zeno's (Xeno's?) Paradox, or "The Tortoise and the Hare." The earliest Greeks couldn't conceive of an INFINITE number of arithmetic operations converging to a FINITE result; hence the "paradox." But in YOUR problem there clearly ARE times beyond 2t and distances beyond D, so the fact that you CHOOSE to divide something up so that you'll need an infinite number of operations to reconstiute it, is neither here nor there, in the end. Similarly, the the Hare always DOES catch and then overtake the Tortoise, despite the way in which the contest is analysed in the standard classical "Paradox."

Note that I said the "earliest" Greeks, above, had difficulties with imagining an infinite number of operations reaching a conclusion in a finite time. But I'm sure that wasn't true of later Greeks, still in the classical Greek era. We can be pretty sure that Archimedes, for example, would have had no trouble with understanding Xeno's Paradox. After all, Archimedes himself essentially obtained the surface area of a sphere by dividing it up into an infinite number of parts and summing them.

Live long and prosper.

Answer 3

If you cover half the distance between you and point B with each move, you will never get there. this is because you will never finish the move. If your initial distance is 1 meter, your first movement is 50 centimeters. Now your remaining distance is 50 centimeters, but you're only going half that so you'll move 25 centimeters. Now you are 25 centimeters away, but will only move 12.5 centimeters. So the distance you cover gets smaller and smaller with each subsequent move.

Answer 4

You will never get there. Every time you move, you only get half way to the target. You will keep getting nearer all the time but never quite getting there.